2 Data used in the study

Table 1 lists the eight QSOs analyzed in this paper. Seven QSOs were observed with the UVES spectrograph at the VLT Kueyen telescope (built by ESO, P.I. S. D'Odorico). The UVES data were reduced with the MIDAS ECHELLE/UVES package. The final reduced vacuum heliocentric spectra have S/N of 30-50 per pixel in the regions of interest and a resolution of $R \sim 45\,000$. The spectra were normalized locally using a 5th order polynomial fit. The normalized spectra were then fitted with Voigt profiles using VPFIT (Carswell et al.: http://www.ast.cam.ac.uk/~rfc/vpfit.html) with the reduced $\chi^{2}$ threshold of 1.3 to obtain the three line parameters, z, b and $N_{\mbox{H~{\sc i}}}$. The metal lines were identified and removed as described in Kim et al. (2001a). Details of the observations and data reduction, and the fitted line lists may be found in Kim et al. (2001a, 2001b).

The line parameters of Q0000-263 were taken from Lu et al. (1996) to include the highest redshift Ly$\alpha$ forest available in the literature, with similar resolution and S/N to the UVES data. Their analysis of Q0000-263 was also undertaken with VPFIT.

In order to avoid confusion with the Ly$\beta$ forest and the proximity effect, we consider only the wavelength range from the Ly$\beta$ emission to 3000 km s^-1 shortward of the Ly$\alpha$ emission. However, the redshift intervals actually used are further limited by other factors such as the incomplete coverage of the forest region, a damped Ly$\alpha$ system and our attempt to overlap the wavelengths of each QSO as much as possible to study the cosmic variance of $b_{\rm c}(N_{\mbox{H~{\sc i}}})$. Table 1 lists the wavelength ranges used for each QSO.

We restrict our analysis to $N_{\mbox{H~{\sc i}}} = 10^{12.5-14.5} {\rm cm}^{-2}$. The lower limit corresponds to the detection threshold in the regions of poorest S/N and the upper limit is where the $N_{\mbox{H~{\sc i}}}$ estimate from fitting Ly$\alpha$alone becomes unreliable because of line saturation. Because lines in blends can also have large uncertainties, we have further restricted the analysis to include only those lines with profile fitting errors less than 25% in $N_{\mbox{H~{\sc i}}}$and b to better define the lower cutoff envelopes (Schaye et al. 2000; Kim et al. 2001a).

In this study, Sample A defines all the lines available from all QSOs which have treated as a single dataset at each z. We also define Sample B in order to study a fluctuation of the Doppler cutoff at similar redshifts. The spectral coverage for each QSO from the same z bin is not uniform. For those QSOs with more than $\sim$600 Å coverage (HE1122-1648, HE2217-2818, Q0055-269 and Q0000-263), the line lists have been divided into two subsets: a group at higher redshifts and a group at lower redshifts. The rest of the QSOs do not have enough coverage to make this splitting possible and provide only one group each. We label the ensemble of these groups Sample B. Each group of Sample B spans 300 Å-350 Å and is defined to have roughly a similar redshift coverage.

Figure 1: The $N_{\mbox{H~{\sc i}}}$-b diagrams at $<z> \, = 2.1$, 3.3 and 3.8. Errors are not displayed. Triangles indicate possible metal lines or lines in the H  I complex mixed with metal lines. Dotted lines indicate the $N_{\mbox{H~{\sc i}}}$ ranges considered in the study. Dot-dashed lines represent the lower $N_{\mbox{H~{\sc i}}}$ fitting threshold actually used in the fit, above which incompleteness is negligible. Solid lines and dashed lines represent the iterative power-law fits and the smoothed b power-law fits, respectively. Shaded area represents a $N_{\mbox{H~{\sc i}}}$-b distribution enclosed by two power-law fits at $<z> \, = 2.1$.